scholarly journals On nonmeasurable selectors of countable group actions

2009 ◽  
Vol 202 (3) ◽  
pp. 281-294 ◽  
Author(s):  
Piotr Zakrzewski
2001 ◽  
Vol 130 (3) ◽  
pp. 383-400 ◽  
Author(s):  
BRIAN H. BOWDITCH ◽  
JOHN CRISP

In this paper we consider group actions on generalized treelike structures (termed ‘pretrees’) defined simply in terms of betweenness relations. Using a result of Levitt, we show that if a countable group admits an archimedean action on a median pretree, then it admits an action by isometries on an ℝ-tree. Thus the theory of isometric actions on ℝ-trees may be extended to a more general setting where it merges naturally with the theory of right-orderable groups. This approach has application also to the study of convergence group actions on continua.


1981 ◽  
Vol 1 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Klaus Schmidt

AbstractThis paper discusses the relations between the following properties o finite measure preserving ergodic actions of a countable group G: strong ergodicity (i.e. the non-existence of almost invariant sets), uniqueness of G-invariant means on the measure space carrying the group action, and certain cohomological properties. Using these properties one can characterize all actions of amenable groups and of groups with Kazhdan's property T. For groups which fall in between these two definations these notions lead to some interesting examples.


2019 ◽  
Vol 40 (10) ◽  
pp. 2593-2680 ◽  
Author(s):  
LEWIS BOWEN

Kolmogorov–Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.


2019 ◽  
Vol 20 (01) ◽  
pp. 2050006
Author(s):  
Cao Zhao ◽  
Yong Ji

In this paper, the mean values of the recurrence are computed for general group actions. Let [Formula: see text] be a metric space with a finite measure [Formula: see text] and [Formula: see text] be a countable group acting on [Formula: see text]. Let [Formula: see text] be a sequence of subsets of [Formula: see text] with [Formula: see text] and put [Formula: see text]. If the Hausdorff measure [Formula: see text] is finite on [Formula: see text] and [Formula: see text] is [Formula: see text]-invariant. We assume that [Formula: see text] and [Formula: see text] are concordant. Then the function [Formula: see text] is [Formula: see text]-integrable and for any [Formula: see text]-measurable set [Formula: see text] we have [Formula: see text] If moreover, [Formula: see text] then [Formula: see text] without the concordance condition for the measure [Formula: see text] and [Formula: see text]


2014 ◽  
Vol 218 (5) ◽  
pp. 777-783
Author(s):  
Darryl McCullough
Keyword(s):  

Author(s):  
Cristina Bertone ◽  
Francesca Cioffi

AbstractGiven a finite order ideal $${\mathcal {O}}$$ O in the polynomial ring $$K[x_1,\ldots , x_n]$$ K [ x 1 , … , x n ] over a field K, let $$\partial {\mathcal {O}}$$ ∂ O be the border of $${\mathcal {O}}$$ O and $${\mathcal {P}}_{\mathcal {O}}$$ P O the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$ O . In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$ ∂ O -marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$ ∂ O -marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $$\partial {\mathcal {O}}$$ ∂ O -marked bases and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.


2019 ◽  
Vol 2019 (753) ◽  
pp. 23-56 ◽  
Author(s):  
Christian Miebach ◽  
Karl Oeljeklaus

AbstractWe systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori’s well-known construction. This yields new examples of non-Kähler compact complex manifolds having free fundamental groups. We then investigate their analytic and geometric invariants such as the Kodaira and algebraic dimension, the Picard group and the deformation theory, thus extending results due to Lárusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky construction allows to recover examples of equivariant compactifications of {{\rm{SL}}(2,\mathbb{C})/\Gamma} for Γ a discrete free loxodromic subgroup of {{\rm{SL}}(2,\mathbb{C})}, previously obtained by A. Guillot.


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